Convergence and accuracy of three finite difference schemes for a two-dimensional conduction and convection problem (original) (raw)
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International Journal for Numerical Methods in Fluids, 1986
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Computer Methods in Applied Mechanics and Engineering, 1982
It is helpful to consider the error in a finite difference solution as arising from two sources: the profile error component arises because the assumed profile used in deriving the scheme does not match the exact solution; the operator error component is due to the failure of the finite-difference operator to accurately simulate the convection-diffusion process. Profile error is relatively easy to quantify but to understand the performance of any differencing scheme to be used in a problem where convection is important, or dominant, requires a test for operator error. The first contribution of this paper is to propose such a test. Upstream difference, central difference, and influence schemes are subjected to the operator error test.
Comparative criteria for finite-difference formulations for problems of fluid flow
International Journal for Numerical Methods in Engineering, 1977
This paper concerns a technique for providing quantitative and qualitative answers to the questions related to accuracy and stability of finite-difference schemes. It is applicable to both the unsteady and the steady flows. The application of the technique provides comparative information about the amplitudes and the speeds of propagation of the numerical and analytic solutions. The difference between the two solutions is characterized in terms of a 'false' propagation speed and 'false' diffusion parameters for the numerical schemes. The technique is applied to a number of commonly used finite-difference schemes and it is concluded that the use of central differences for the convective terms and/or explicit formulations tends to increase the amplitudes and wave speeds. The opposite effects on the amplitudes and wave speeds are produced by upwind or 'donor' cell differences and/or implicit formulations.
Advances in Difference Equations, 2017
We discuss a new nine-point fourth-order and five-point second-order accurate finite-difference scheme for the numerical solution of two-space dimensional convection-diffusion problems. The compact operators are defined on a quasi-variable mesh network with the same order and accuracy as obtained by the central difference and averaging operators on uniform meshes. Subsequently, a high-order difference scheme is developed to get the numerical accuracy of order four on quasi-variable meshes as well as on uniform meshes. The error analysis of the fourth-order compact scheme is described in detail by means of matrix analysis. Some examples related with convection-diffusion equations are provided to present performance and robustness of the proposed scheme.
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Applied Mathematical Modelling, 1985
False-diffusion errors in numerical solutions of convection-diffusion problems, in two-and three-dimensions, arise from the numerical approximations of the convection term in the conservation equations. For finite difference-based methods, one way to overcome these errors is to use an upwind approximation which essentially follows the streamlines. This approach, originally derived by Raithby is formally called the skewupwind differencing scheme. Although this scheme shows promise and has proved to be more accurate than most others on a limited number of test problems, it does have some shortcomings. The method outlined in this paper retains the general objectives of the Raithby approach, but uses an entirely different formulation that eliminates the shortcomings of the original scheme. The paper describes the formulation of the proposed method, and demonstrates its performance on a standard test problem.
A monotone finite-difference high order accuracy scheme for the 2D convection – diffusion equations
Journal of the Belarusian State University. Mathematics and Informatics
A stable finite-difference scheme is constructed on a minimum stencil of a uniform mesh for a two-dimensional steady-state convection – diffusion equation of a general form; the scheme is theoretically studied and tested. It satisfies the maximum principle and has the fourth order of approximation. The scheme monotonicity is controlled by two regularization parameters introduced into the difference operator. The scheme is focused on solving applied convection – diffusion problems with a developed boundary layer, including gravitational convection, thermomagnetic convection, and diffusion of particles in a magnetic fluid. The scheme is tested on the well-known problem of a high-intensive gravitational convection in a horizontal channel of a square cross-section with a uniform heating from the side. A detailed comparison is performed with the monotone Samarskii scheme of the second order approximation on the sequences of square meshes with the number of partitions from 10 to 1000 on e...
International Journal for Numerical Methods in Fluids, 2009
This article deals with the study of the development and application of the high-order upwind ADBQUICKEST scheme, an adaptative bounded version of the QUICKEST for unsteady problems (Commun. Numer. Meth. Engng 2007; 23:419-445), employing both linear and nonlinear convection term discretization. This scheme is applicable to a wide range of computational fluid dynamics problems, where transport phenomena are of special importance. In particular, the performance of the scheme is assessed through an extensive numerical simulation study of advection-diffusion problems. The scheme, implemented in the context of finite difference methodology, combines a good approximation of shocks (or discontinuities) with a good approximation of the smooth parts of the solutions. In order to assess the performance of the scheme, seven problems are solved, namely (a) advection of scalars; (b) non-linear viscous Burgers equation; (c) Euler equations of gas dynamics; (d) Newtonian flow in a channel; (e) axisymmetric Newtonian jet flow; (f) axisymmetric non-Newtonian (generalized Newtonian) flow in a pipe; and (g) collapse of a fluid column. The numerical experiments clearly show that the scheme provides more consistent solutions than those found in the literature. From the study, the flexibility and robustness of the ADBQUICKEST scheme is confirmed by demonstrating its capability to solve a variety of linear and nonlinear problems with and without discontinuous solutions.
The upwind hybrid difference methods for a convection diffusion equation
Applied Numerical Mathematics, 2018
We propose the upwind hybrid difference method and its penalized version for the convection dominated diffusion equation. The hybrid difference method is composed of two types of approximations: one is the finite difference approximation of PDEs within cells (cell FD) and the other is the interface finite difference (interface FD) on edges of cells. The interface finite difference is derived from continuity of normal fluxes. The penalty method is obtained by adding small diffusion in the interface FD. The penalty term makes it possible to reduce severe numerical oscillations in the upwind hybrid difference solutions. The penalty parameter is designed to be some power of the grid size. A complete stability is provided. Convergence estimates seems to be conservative according to our numerical experiments. To exposit convergence property and controllability of numerical oscillations several numerical tests are provided.
Analysis of finite element schemes for convection-type problems
International Journal for Numerical Methods in Fluids, 1995
Various finite element schemes of the Bubnov–Galerkin and Taylor–Galerkin types are analysed to obtain the expressions of truncation errors. This way, dispersion errors in the transient, and diffusion errors both in the transient and in the steady state, are identified. Then, with reference to the transient advection–diffusion equation, stability limits are determined by means of a general von Neumann procedure. Finally, the operational equivalence between Taylor–Galerkin methods, utilized for pseudo-transient calculations, and Petrov–Galerkin methods, derived for the steady state forms of the advection–diffusion equation, is illustrated. Theoretical conclusions are supported by the results of numerical experiments.